In certain sense the question of finding qKdV spectra can be regarded as solved: there is not one but two distinct ways to write an algebraic Bethe-ansatz reducing the problem of finding spectra to a bunch of algebraic equations [1]}, [2]}. In practice complexity of these equations grows very rapidly with the level \(m\)  (REF ), making this approach useless in the context of ETH, at least so far. The ETH holds in thermodynamic limit, it may not and does not hold beyond that regime. Thermodynamic limit assumes the length of the spatial circle \(L\)  goes to infinity, with the energy density \(E/L\)  kept fixed. Using rescaling, one can always bring the circle to unit radius, the notations we use throughout the paper. The energy \(E\)  then must go to infinity as \(L^2\)  with \(L\rightarrow \infty \)  being an auxiliary parameter keeping track of corrections to various ETH-related identities. For any given primary state \(|\Delta \rangle \)  this essentially means the descendant level \(m\)  must be taken to infinity, i.e. we arrive exactly at the limit where algebraic Bethe equations become most difficult.
